I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side...

Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem ϕ(x) → min, x ∈ ℝ m , are given. These conditions work with arbitrary functions ϕ: ℝ m → ℝ, but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives. It is shown that the answer is affirmative if ϕ is of class $$\mathcal{C}^{1,1}$$ (i.e., differentiable with locally Lipschitz derivative).Further, considering $$\mathcal{C}^{1,1}$$ functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance” from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions given by Liu, Neittaanmaki, Krizek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient not only for efficiency but also for strict efficiency.

Let A: V → V′ be a strongly elliptic operator on a d-dimensional manifold D (polyhedra or boundaries of polyhedra are also allowed). An operator equation Au = f with stochastic data f is considered. The goal of the computation is the mean field and higher moments $$\mathcal{M}^1 u \in V,\mathcal{M}^2 u \in V \otimes V,...,\mathcal{M}^k u \in V \otimes ... \otimes V$$ of the solution.We discretize the mean field problem using a FEM with hierarchical basis and N degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment $$\mathcal{M}^k u$$ for k⩾1.The key tool in both algorithms is a “sparse tensor product” space for the approximation of $$\mathcal{M}^k u$$ with O(N(log N) k−1) degrees of freedom, instead of N k degrees of freedom for the full tensor product FEM space.A sparse Monte-Carlo FEM with M samples (i.e., deterministic solver) is proved to yield approximations to $$\mathcal{M}^k u$$ with a work of O(M N(log N) k−1) operations. The solutions are shown to converge with the optimal rates with respect to the Finite Element degrees of freedom N and the number M of samples.The deterministic FEM is based on deterministic equations for $$\mathcal{M}^k u$$ in D k ⊂ ℝkd. Their Galerkin approximation using sparse tensor products of the FE spaces in D allows approximation of $$\mathcal{M}^k u$$ with O(N(log N) k−1) degrees of freedom converging at an optimal rate (up to logs).For nonlocal operators wavelet compression of the operators is used. The linear systems are solved iteratively with multilevel preconditioning. This yields an approximation for $$\mathcal{M}^k u$$ with at most O(N (log N) k+1) operations.

In this paper we study the q-version of the Partition of Unity Method for the Helmholtz equation. The method is obtained by employing the standard bilinear finite element basis on a mesh of quadrilaterals discretizing the domain as the Partition of Unity used to paste together local bases of special wave-functions employed at the mesh vertices. The main topic of the paper is the comparison of the performance of the method for two choices of local basis functions, namely a) plane-waves, and b) wave-bands. We establish the q-convergence of the method for the class of analytical solutions, with q denoting the number of plane-waves or wave-bands employed at each vertex, for which we get better than exponential convergence for sufficiently small h, the mesh-size of the employed mesh. We also discuss the a-posteriori estimation for any solution quantity of interest and the problem of quadrature for all integrals employed. The goal of the paper is to stimulate theoretical development which could explain various numerical features. A main open question is the analysis of the pollution and its disappearance as function of h and q.

Numerical simulation of turbulent flows is one of the great challenges in Computational Fluid Dynamics (CFD). In general, Direct Numerical Simulation (DNS) is not feasible due to limited computer resources (performance and memory), and the use of a turbulence model becomes necessary. The paper will discuss several aspects of two approaches of turbulent modeling—Large Eddy Simulation (LES) and Variational Multiscale (VMS) models. Topics which will be addressed are the detailed derivation of these models, the analysis of commutation errors in LES models as well as other results from mathematical analysis.

By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported.

A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in L 2(Ω) involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced.

Various types of exponential smoothing for data observed at irregular time intervals are surveyed. Double exponential smoothing and some modifications of Holt’s method for this type of data are suggested. A real data example compares double exponential smoothing and Wright’s modification of Holt’s method for data observed at irregular time intervals.

In this paper, we obtain all possible general solutions of the sum form functional equations $$\sum\limits_{i = 1}^k {\sum\limits_{j = 1}^l {f(p_i q_j )} } = \sum\limits_{i = 1}^k {g(p_i )} \sum\limits_{j = 1}^l {h(q_j )} $$ and $$\sum\limits_{i = 1}^k {\sum\limits_{j = 1}^l {F(p_i q_j )} } = \sum\limits_{i = 1}^k {G(p_i ) + } \sum\limits_{j = 1}^l {H(q_j ) + \lambda } \sum\limits_{i = 1}^k {G(p_i )} \sum\limits_{j = 1}^l {H(q_j )} $$ valid for all complete probability distributions (p 1, ..., p k), (q 1, ..., q l ), k ≥ 3, l ≥ 3 fixed integers; λ ∈ ℝ, λ ≠ 0 and F, G, H, f, g, h are real valued mappings each having the domain I = [0, 1], the unit closed interval.

In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics of the multi-dimensional hyperbolic system. In this way all of the infinitely many directions of wave propagation are taken into account. The main goal of this paper is to present a self-contained overview on the recent results. We study the L 1-stability of the finite volume schemes obtained by various approximations of the flux integrals. Several numerical experiments presented in the last section confirm robustness and correct multi-dimensional behaviour of the FVEG methods.

In the present paper, a mathematical model, originally proposed by Danziger and Elmergreen and describing the thyroid-pituitary homeostatic mechanism, is modified and analyzed for its physiological and clinical significance. The inuence of different system parameters on the stability behavior of the system is discussed. The transportation delays of different hormones in the bloodstream, both in the discrete and distributed forms, are considered. Delayed models are analyzed regarding the stability and bifurcation behavior. Clinical treatment of periodic catatonic schizophrenia is discussed in presence of transportation delays. Numerical simulations are presented to support analytic results.

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L p Young measure theory and related compactness results, in the first section. Then we use the L p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.

For convection-diffusion problems with exponential layers, optimal error estimates for linear finite elements on Shishkin-type meshes are known. We present the first optimal convergence result in an energy norm for a Bakhvalov-type mesh.

Some inequalities for the Stieltjes integral and applications in numerical integration are given. The Stieltjes integral is approximated by the product of the divided difference of the integrator and the Lebesgue integral of the integrand. Bounds on the approximation error are provided. Applications to the Fourier Sine and Cosine transforms on finite intervals are mentioned as well.

We compare the numerical performance of several methods for solving the discrete contact problem arising from the finite element discretisation of elastic systems with numerous contact points. The problem is formulated as a variational inequality and discretised using piecewise quadratic finite elements on a triangulation of the domain. At the discrete level, the variational inequality is reformulated as a classical linear complementarity system. We compare several state-of-art algorithms that have been advocated for such problems. Computational tests illustrate the use of these methods for a large collection of elastic bodies, such as a simplified bidimensional wall made of bricks or stone blocks, deformed under volume and surface forces.

We present a detailed proof of the density of the set $$C^\infty (\bar \Omega ) \cap V$$ in the space of test functions V ⊂ H 1 (Ω) that vanish on some part of the boundary ∂Ω of a bounded domain Ω.

We provide a general series form solution for second-order linear PDE system with constant coefficients and prove a convergence theorem. The equations of three dimensional elastic equilibrium are solved as an example. Another convergence theorem is proved for this particular system. We also consider a possibility to represent solutions in a finite form as partial sums of the series with terms depending on several complex variables.

In the paper we study the equation L u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.

The paper deals with homogenization of a linear elliptic boundary problem with a specific class of uncertain coefficients describing composite materials with periodic structure. Instead of stochastic approach to the problem, we use the worst scenario method due to Hlaváček (method of reliable solution). A few criterion functionals are introduced. We focus on the range of the homogenized coefficients from knowledge of the ranges of individual components in the composite, on the values of generalized gradient in the places where these components change and on the average of homogenized solution in some critical subdomain.